Observation of Different Behaviors of Logistic Map for Different Control Parameters

Musammet Tahmina Akter; Mohammad Abul Mansur Chowdhury

doi:doi:10.11648/j.ijamtp.20180403.14

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Observation of Different Behaviors of Logistic Map for Different Control Parameters

Musammet Tahmina Akter, Mohammad Abul Mansur Chowdhury

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 4, Issue 3)

Received: 3 October 2018 Accepted: 8 November 2018 Published: 4 December 2018

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Abstract

The logistic map is one of the most important but common examples of chaotic dynamics. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. We put an importance on report of the Verhulst logistic map which is one of the potential models and methods for researching dynamical systems that could develop to chaotic. Chaotic signals present a special difficulty in parameter estimation. The difficulty arises from the definition of a chaotic system because of sensitive dependence on initial conditions. It is seen that very slight changes in the initial conditions cause significant effects in the evolution. In general the chaotic systems are nonlinear and apparently random but they are deterministic. The main objective of this paper is how can find the logistic map equation and investigated the chaotic behavior for the logistic equation by varying the control parameters and finally discover Lyaponov exponent, Bifurcation diagrams etc.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 4, Issue 3)
DOI	10.11648/j.ijamtp.20180403.14
Page(s)	84-90
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Logistic Map, Chaos, Lyapunov Exponent, Bifurcation, Cobweb, Attractor

References

[1]	Díaz-Méndez A, MarquinaPérez JV, Cruz-Irrison M, Vázquez-Medina R, Del-Río-Correa JL. Chaotic noise MOS generator based on Logistic map. Micro electronics Journal, 2009, 40: 638-640.
[2]	Siji PD, Rajesh R Takagi-Sugeno Fuzzy modelling of Logistic map using genetic algorithm. International Journal of Wisdom Based Computing, 2011, 1(3): 9-13.
[3]	George M. Stability areas in Logistic map. Advanced Research in Scientific Areas, 2012.
[4]	Wei JG, Leng G. Lyapunov exponent and Chaos of Duffing’s equation perturbed by white noise. Applied Mathematics and Computation, 1997, 88: 77-93.
[5]	Kathira M. A Lyapunov exponent approach for identifying chaotic behaviour in a finiteelement based drills string vibration model. A thesis submitted to the office of Graduate Studies of Texas, A & M University in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering, 2009.
[6]	Shuichi A, Yoshifumi N. A chaotic cryptosystem using Lyapunov exponent. The 15^th IEEE International Workshop on Nonlinear Dynamics of Electronic Systems, 2007, NDE’07Tokushima, Japan.
[7]	Andrzej S, Tomasz K. Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of mass synchronization. Chaos, Solitons and Fractals, 2003; 15: 233-244.
[8]	Andrzej S, Artur D, Tomasz K. Evaluation of the largest Lyapunov exponent indynamical systems with time delay. Chaos, Solitons and Fractals, 2005; 23: 1651-1659.
[9]	T. A. O. Salau and O. O. Ajide, Development of a Lyapunov Exponent Based Chaos Diagram in the Parameter Plane of Logistic Map, British Journal of Applied Science & Technology, 2014, 4(21): 3096- 3106.
[10]	Nandi A. , Dutta D., Bhattacharjee J. K. and Ramaswamy R., 2005, Chaos 15, 023107, DOI: 10.1063/1.1914755.
[11]	M. Tahmina Akter, Studies of chaos in non-linear dynamical systems, a thesis submitted to the research centre for mathematical and physical sciences (RCMPS), University of Chittagong, Chittagong-4331, Bangladesh in partial fulfillment of the requirements for the degree of Master of Philosophy (M. phil.), 2012.

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Musammet Tahmina Akter, Mohammad Abul Mansur Chowdhury. (2018). Observation of Different Behaviors of Logistic Map for Different Control Parameters. International Journal of Applied Mathematics and Theoretical Physics, 4(3), 84-90. https://doi.org/10.11648/j.ijamtp.20180403.14

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ACS Style

Musammet Tahmina Akter; Mohammad Abul Mansur Chowdhury. Observation of Different Behaviors of Logistic Map for Different Control Parameters. Int. J. Appl. Math. Theor. Phys. 2018, 4(3), 84-90. doi: 10.11648/j.ijamtp.20180403.14

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AMA Style

Musammet Tahmina Akter, Mohammad Abul Mansur Chowdhury. Observation of Different Behaviors of Logistic Map for Different Control Parameters. Int J Appl Math Theor Phys. 2018;4(3):84-90. doi: 10.11648/j.ijamtp.20180403.14

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@article{10.11648/j.ijamtp.20180403.14,
  author = {Musammet Tahmina Akter and Mohammad Abul Mansur Chowdhury},
  title = {Observation of Different Behaviors of Logistic Map for Different Control Parameters},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {4},
  number = {3},
  pages = {84-90},
  doi = {10.11648/j.ijamtp.20180403.14},
  url = {https://doi.org/10.11648/j.ijamtp.20180403.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20180403.14},
  abstract = {The logistic map is one of the most important but common examples of chaotic dynamics. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. We put an importance on report of the Verhulst logistic map which is one of the potential models and methods for researching dynamical systems that could develop to chaotic. Chaotic signals present a special difficulty in parameter estimation. The difficulty arises from the definition of a chaotic system because of sensitive dependence on initial conditions. It is seen that very slight changes in the initial conditions cause significant effects in the evolution. In general the chaotic systems are nonlinear and apparently random but they are deterministic. The main objective of this paper is how can find the logistic map equation and investigated the chaotic behavior for the logistic equation by varying the control parameters and finally discover Lyaponov exponent, Bifurcation diagrams etc.},
 year = {2018}
}

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TY  - JOUR
T1  - Observation of Different Behaviors of Logistic Map for Different Control Parameters
AU  - Musammet Tahmina Akter
AU  - Mohammad Abul Mansur Chowdhury
Y1  - 2018/12/04
PY  - 2018
N1  - https://doi.org/10.11648/j.ijamtp.20180403.14
DO  - 10.11648/j.ijamtp.20180403.14
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 84
EP  - 90
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20180403.14
AB  - The logistic map is one of the most important but common examples of chaotic dynamics. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. We put an importance on report of the Verhulst logistic map which is one of the potential models and methods for researching dynamical systems that could develop to chaotic. Chaotic signals present a special difficulty in parameter estimation. The difficulty arises from the definition of a chaotic system because of sensitive dependence on initial conditions. It is seen that very slight changes in the initial conditions cause significant effects in the evolution. In general the chaotic systems are nonlinear and apparently random but they are deterministic. The main objective of this paper is how can find the logistic map equation and investigated the chaotic behavior for the logistic equation by varying the control parameters and finally discover Lyaponov exponent, Bifurcation diagrams etc.
VL  - 4
IS  - 3
ER  -

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Author Information

Musammet Tahmina Akter

Department of Mathematics, Chittagong University of Engineering & Technology, Chittagong, Bangladesh
Mohammad Abul Mansur Chowdhury

Jamal Nazrul Islam Research Center for Mathematical and Physical Sciences, University of Chittagong, Chittagong, Bangladesh

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Plain Text BibTeX RIS

APA Style

Musammet Tahmina Akter, Mohammad Abul Mansur Chowdhury. (2018). Observation of Different Behaviors of Logistic Map for Different Control Parameters. International Journal of Applied Mathematics and Theoretical Physics, 4(3), 84-90. https://doi.org/10.11648/j.ijamtp.20180403.14

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ACS Style

Musammet Tahmina Akter; Mohammad Abul Mansur Chowdhury. Observation of Different Behaviors of Logistic Map for Different Control Parameters. Int. J. Appl. Math. Theor. Phys. 2018, 4(3), 84-90. doi: 10.11648/j.ijamtp.20180403.14

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AMA Style

Musammet Tahmina Akter, Mohammad Abul Mansur Chowdhury. Observation of Different Behaviors of Logistic Map for Different Control Parameters. Int J Appl Math Theor Phys. 2018;4(3):84-90. doi: 10.11648/j.ijamtp.20180403.14

Copy | Download

@article{10.11648/j.ijamtp.20180403.14,
  author = {Musammet Tahmina Akter and Mohammad Abul Mansur Chowdhury},
  title = {Observation of Different Behaviors of Logistic Map for Different Control Parameters},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {4},
  number = {3},
  pages = {84-90},
  doi = {10.11648/j.ijamtp.20180403.14},
  url = {https://doi.org/10.11648/j.ijamtp.20180403.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20180403.14},
  abstract = {The logistic map is one of the most important but common examples of chaotic dynamics. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. We put an importance on report of the Verhulst logistic map which is one of the potential models and methods for researching dynamical systems that could develop to chaotic. Chaotic signals present a special difficulty in parameter estimation. The difficulty arises from the definition of a chaotic system because of sensitive dependence on initial conditions. It is seen that very slight changes in the initial conditions cause significant effects in the evolution. In general the chaotic systems are nonlinear and apparently random but they are deterministic. The main objective of this paper is how can find the logistic map equation and investigated the chaotic behavior for the logistic equation by varying the control parameters and finally discover Lyaponov exponent, Bifurcation diagrams etc.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Observation of Different Behaviors of Logistic Map for Different Control Parameters
AU  - Musammet Tahmina Akter
AU  - Mohammad Abul Mansur Chowdhury
Y1  - 2018/12/04
PY  - 2018
N1  - https://doi.org/10.11648/j.ijamtp.20180403.14
DO  - 10.11648/j.ijamtp.20180403.14
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 84
EP  - 90
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20180403.14
AB  - The logistic map is one of the most important but common examples of chaotic dynamics. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. We put an importance on report of the Verhulst logistic map which is one of the potential models and methods for researching dynamical systems that could develop to chaotic. Chaotic signals present a special difficulty in parameter estimation. The difficulty arises from the definition of a chaotic system because of sensitive dependence on initial conditions. It is seen that very slight changes in the initial conditions cause significant effects in the evolution. In general the chaotic systems are nonlinear and apparently random but they are deterministic. The main objective of this paper is how can find the logistic map equation and investigated the chaotic behavior for the logistic equation by varying the control parameters and finally discover Lyaponov exponent, Bifurcation diagrams etc.
VL  - 4
IS  - 3
ER  -

Copy | Download